# Dynamical systems

My research area is within dynamical systems — a branch of mathematics which strongly interacts with more classical disciplines, such as geometry, topology, (numerical) analysis, and measure and probability theory. The general aim is to understand the long-term behaviour of nonlinear deterministic systems and qualitative changes in dynamics upon variation of system parameters. The transition from orderly to complex chaotic dynamics is particularly important. Rather than studying individual evolutions, the goal is to obtain a global and qualitative overview of the dynamics by studying the geometric organisation of the product of state and parameter spaces.

## Publications

**Pelzer, A.F.G. and Sterk, A.E.**(2020)*Finite cascades of pitchfork bifurcations and multistability in generalized Lorenz-96 models.*. . (URL) (BibTeX)**De Jong, T.G., Sterk, A.E. and Broer, H.W.**(2020)*Fungal tip growth arising through a codimension-1 global bifurcation.*. . (URL) (BibTeX)**Gaiko, V.A., Broer, H.W. and Sterk, A.E.**(2019)*Global bifurcation analysis of Topp system.*. . (URL) (BibTeX)**De Jong, T.G., Sterk, A.E. and Guo, F.**(2019)*Numerical method to compute hypha tip growth for data driven validation.*. . (URL) (BibTeX)**Van Kekem, D.L. and Sterk, A.E.**(2019)*Symmetries in the Lorenz-96 model.*. . (URL) (BibTeX)**Ghane, H., Sterk, A.E. and Waalkens, H.**(2019)*Chaotic dynamics from a pseudo-linear system.*. . (URL) (BibTeX)**Van Kekem, D.L. and Sterk, A.E.**(2018)*Wave propagation in the Lorenz-96 model.*. . (URL) (BibTeX)**Garst, S. and Sterk, A.E.**(2018)*Periodicity and chaos amidst twisting and folding in 2-dimensional maps.*. . Feature article. (URL) (BibTeX)**Van Kekem, D.L. and Sterk, A.E.**(2018)*Travelling waves and their bifurcations in the Lorenz-96 model.*. . (URL) (BibTeX)**Garst, S. and Sterk, A.E.**(2016)*The dynamics of a fold-and-twist map.*. . (URL) (BibTeX)**Maathuis, Henry, Boulogne, Luuk, Wiering, Marco and Sterk, Alef**(2017)*Predicting chaotic time series using machine learning techniques.*In Preproceedings of the 29th Benelux Conference on Artificial Intelligence (BNAIC 2017). November. (Bart Verheij and Marco Wiering, Eds.) University of Groningen, pages 326-340. (URL) (BibTeX)

# Low-frequency climate variability

I have used dynamical systems theory to study low-frequency variability in atmosphere and ocean models. The models are systems of partial differential equations. Using Galerkin projections I derived systems of ordinary differential equations, up to 73 degrees of freedom, modeling the low-frequency dynamics. To study the dynamics of the reduced models I used the C programming language to develop dedicated software implementing numerical tools, such as integration and continuation algorithms, Lyapunov exponents, and power spectra.

In Sterk et al. (2010) we study the dynamics of a reduced atmosphere model derived from the 2-layer shallow water equations. Recurrent flow patterns manifest themselves as strange attractors appearing after successive bifurcations of a periodic attractor. These attractors represent irregular planetary waves which "inherit" their spatio-temporal characteristics from the periodic attractor. This scenario differs from those explaining recurrent flow patterns in terms of intermittency near multiple equilibria.

In Broer et al. (2011) we study the dynamics of a reduced ocean model derived from the primitive equations. This model has a periodic attractor with the spatio-temporal characteristics of the Atlantic Multidecadal Oscillation, which is a distinct signal of temperature variability in the North Atlantic ocean. Period doubling cascades and annual heat flux forcing give rise to (quasi-periodic) Henon-like strange attractors.

## Publications

**Broer, H.W., Dijkstra, H.A., Sim\'o, C., Sterk, A.E. and Vitolo, R.**(2011)*The dynamics of a low-order model for the Atlantic Multidecadal Oscillation.*. . (URL) (BibTeX)**Sterk, A.E., Vitolo, R., Broer, H.W., Sim\'o, C. and Dijkstra, H.A.**(2010)*New nonlinear mechanisms of midlatitude atmospheric low-frequency variability.*. . (URL) (BibTeX)**Sterk, A.E.**(2010)*Atmospheric Variability and the Atlantic Multidecadal Oscillation: Mathematical Analysis of Low-Order Models.*PhD thesis. (URL) (BibTeX)

# Extreme events

Classical extreme value theory studies the distributions of large values in time series of independent, identically distributed random variables. Since the early 2000s the theory has been extended to deterministic dynamical systems. In this setting one studies the extremes of a time series obtained by evaluating a scalar observable along an evolution of the system.

The so-called Generalised Extreme Value (GEV) distribution can be used to compute the probability of occurrence of future large values of a quantity, given a sample of past measurements. The GEV depends on three parameters. Of particular importance is the so-called tail index because it determines the tail width of the distribution, and, therefore, the likelihood of extreme values. The tail index is typically estimated from long-term simulations. Recently, we discovered that in certain classes of deterministic systems the tail index is related to the dimension of the attractor of the underlying dynamical system (Holland et al., 2012).

Also the predictability of extreme events is relevant. Predictability is typically measured in terms of the growth rate of errors in the initial conditions. In Sterk et al. (2012) we have used finite-time Lyapunov exponents to quantify the predictability of extreme values, such as convection, energy, and wind speeds, in geophysical models. In Sterk et al. (2015) we have studied the predictability of extreme wind speeds using ensemble forecasts produced by the MOGREPS system of the Met Office.

## Publications

**Holland, M.P. and Sterk, A.E.**(2021)*On max-semistable laws and extremes for dynamical systems.*. . (URL) (BibTeX)**Boer, N.B. and Sterk, A.E.**(2021)*Generalized Fibonacci numbers and extreme value laws for the R\'enyi map.*. . (URL) (BibTeX)**Sterk, A.E. and M.P. Holland**(2018)*Extreme value laws and mean squared error growth in dynamical systems.*. . (URL) (BibTeX)**Sterk, A.E. and Van Kekem, D.L.**(2017)*Predictability of extreme waves in the Lorenz-96 model near intermittency and quasi-periodicity.*. . (URL) (BibTeX)**Sterk, A.E.**(2016)*Extreme amplitudes of a periodically forced Duffing oscillator.*. . (URL) (BibTeX)**Holland, M.P., Rabassa, P. and Sterk, A.E.**(2016)*Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems.*. . (URL) (BibTeX)**Sterk, A.E., Stephenson, D.B., Holland, M.P. and Mylne, K.R.**(2016)*On the predictability of extremes: does the butterfly effect ever decrease?*. . (URL) (BibTeX)**Sterk, A.E., Holland, M.P., Rabassa, P., Broer, H.W. and Vitolo, R.**(2012)*Predictability of extreme values in geophysical models.*. . (URL) (BibTeX)**Holland, M.P., Vitolo, R., Rabassa, P., Sterk, A.E. and Broer, H.W.**(2012)*Extreme value laws in dynamical systems under physical observables.*. . (URL) (BibTeX)